Convergence of the Nash System

This chapter talks about addressing the convergence problem, which is devoted to the convergence of the Nash system. It contains several results on the differential calculus on the space of probability measures together with an Itô's formula for functionals of a process taking values in the space of probability measures. For simplicity, most of the analysis provided in the chapter is on the torus, but the method is robust enough to accommodate the nonperiodic setting. The chapter also shows that monotonicity plays no role in the proofs of certain theorems. Basically, only the global Lipschitz properties of H and DpH, together with the nondegeneracy of the diffusions and the various bounds obtained for the solution of the master equation and its derivatives matter.